2
$\begingroup$

I am trying to reduce a symbolic matrix expression containing augmented/concatenated matrices using TensorReduce, but it does not behave as expected when transposing an augmented matrix.

I assume the problem is that building a matrix from multiple submatrices creates a tensor of higher dimension instead of a bigger 2d tensor, but I do not know how to create a symbolic 2d tensor from multiple smaller symbolic 2d tensors.

As an example

$Assumptions = {
    Element[P, Matrices[{n, n}]]
};
TensorReduce[Transpose[{{P, P}}]]

yields

{{P}, {P}}

and not

{{Transpose[P]}, {Transpose[P]}}

which is the result I would expect.

$\endgroup$
1
$\begingroup$

The tensor functionality will not produce a mixed output of explicit and symbolic matrices. That is, the tensor will remain symbolic unless all of the parts are explicit matrices/arrays. Also, it is better to use TensorTranspose instead of Transpose when dealing with tensors. At any rate, we can write your expression as both a mixed expression and as a tensor expression as follows:

mixed = {{P, P}};
tensor = TensorProduct[{{1, 1}}, P];

Let's check that the two are equivalent by substituting an explicit matrix for P:

mixed == tensor /. P->Array[a, {3,3}]

True

Now, the tensor framework, and in particular TensorTranspose, does not know how to deal with this mixed form. For example:

$Assumptions = P ∈ Arrays[{n, n}];
TensorTranspose[{{P, P}}, {1, 2, 4, 3}]

TensorTranspose::lowlen: Required length 2 is smaller than maximum 4 of support of {1,2,4,3}.

TensorTranspose[{{P, P}}, {1, 2, 4, 3}]

On the other hand, TensorTranspose (as well as Transpose) knows how to deal with the tensor form:

transpose = TensorReduce @ TensorTranspose[tensor, {1, 2, 4, 3}]

TensorTranspose[P \[TensorProduct] {{1, 1}}, {4, 3, 1, 2}]

Let's check that this agrees with your expected result when substituting an explicit matrix for P:

expected = {{Transpose[P], Transpose[P]}};
expected == transpose /. P->Array[a, {3,3}]

True

Summarizing, if you use:

TensorProduct[{{1, 1}}, P]

instead of:

{{P, P}}

then the tensor framework is more likely able to do some transformations of the input.

| improve this answer | |
$\endgroup$
0
$\begingroup$

What you are expecting does not happen for a number of reasons. Firstly, I think, Transpose operates on the assumption that the elements of the list are scalars, regardless of any assumption you give.

Assuming[Element[P, Matrices[{n, n}]], Transpose[{{P, P}}]]
(* {{P}, {P}} *)

TensorReduce has nothing left to do here.

Further, according to the documentation, Transpose "transposes the first two levels in list". This means that

 With[{P = Array[a, {2, 2}]}, Print[P]; Transpose[{{P, P}}]]

(* {{a[1,1],a[1,2]},{a[2,1],a[2,2]}} *)
(* {{{{a[1, 1], a[1, 2]}, {a[2, 1], a[2, 2]}}}, {{{a[1, 1], 
    a[1, 2]}, {a[2, 1], a[2, 2]}}}} *)

So the matrices inside do not get transposed.

| improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.